In heat exchanger cores such as motor vehicle radiator cores, such cores endure many thermal cycles in their service life time as engine coolant temperatures swing between ambient and operating temperatures. Tube metal temperatures follow the coolant temperatures and as a result, the tubes connected to the headers grow/contract axially to a substantial degree. On the other hand, the channel-shaped core reinforcements which are also normally connected to the headers are far less affected by the coolant temperatures and therefore grow at a different rate. As a result, the core reinforcements induce stress in the tube-to-header joints during the coolant thermal cycles that can be significant. Heretofore, such thermally induced tube-to-header joint stress has been relieved where need be by saw cutting the reinforcements following brazing and before the radiator is placed in service. However, the saw cutting operation is a difficult process to automate because of the difficulties associated with handling cutting lubricants and metal chips. Moreover, saw blade maintenance, air center destruction immediately adjacent to the saw cut and the potential for damaging the tube nearest the reinforcement makes the saw cutting operation undesirable.